Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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A Sylow $2-$subgroup of SL(2,q) is irreducible

Let $q$ an odd prime power. By a classic result, a Sylow $2-$subgroup of $SL(2,q)$ is generalized quaternion. How can I show that it is an (absolutely) irreducible subgroup of $GL(2,q)$ ? I have ...
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26 views

Permutation representation for low degree

Thanks for any answer. Suppose $n\leq 10$ and $n\neq 6$ and $k\geq 3$. How can I find all faithful permutation representation of $S_k$ in $S_n$? I mean is there any faithful representation except ...
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2answers
116 views

Conjugates and commutators for twisty puzzles — so what?

This question isn't just rhetorical. I want to know what I'm missing. Twisty puzzle tutorials keep talking about how useful conjugates (operation sequences of the form ${XYX}^{-1}$) and commutators ($...
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57 views

All but a finite number of finite simple groups are groups of matrices over $\mathbb{F}_q$

In the introduction to this honors thesis, http://people.math.gatech.edu/~jrabinoff6/papers/building.pdf I found this statement: Matrix groups defined over the finite fields $\mathbb{F}_q$ ...
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1answer
43 views

Order of subgroup generated by two cyclic subgroups in $S_6$.

Let $S_6$ be the symmetric group, and $\alpha=(13456)$ and $\beta=(132)$ be its two permutations. How can we find the order of the subgroup generated by $\alpha$ and $\beta$. SOl: $\alpha^5$=...
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32 views

Properties of non-abelian characters

I'm looking for some (short of) non-abelian generalization of the following result: Let $G$ be a finite abelian group and let $f$ be a function on $G$ with values in some field of characteristic ...
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1answer
25 views

Groups with order a product of unrelated distinct primes

Consider a group of size $n$, where $n$ is the product of distinct unrelated primes (two primes $p$ and $q$ are unrelated if $q \nmid (p-1)$ and $p \nmid (q-1)$). The claim is that there is only one ...
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1answer
60 views

If $|G/H|=4$ then $G$ is union of three proper subgroups

Let $G$ be a finite group and $N$ a normal subgroup of $G$ of index $4$. Prove that $G$ has a subgroup of index $2$ (hence normal) and if $G/H$ is not cyclic then $G$ is equal to the union of three ...
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62 views

Abelian groups of order 63

I am trying to learn abstract algebra on my own. Unfortunately I am confused and not sure how to proceed with the following question. I want to find all abelian groups of order 63. By theorem of ...
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1answer
65 views

“Concrete” Realisations of non-abelian finite groups

Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are ...
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1answer
53 views

Proof of Lagrange's four square theorem using Cauchy-Davenport Theorem

Cauchy used the Cauchy-Davenport theorem to prove that $ax^2 + by^2 + c \equiv 0 \pmod p$ has solutions provided that $abc \neq 0$. Lagrange used this result to establish his four squares theorem. I ...
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3answers
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Sylow $p$-subgroup of a finite group

I know: Let $P$ be a Sylow $p$-subgroup of a finite group $G$.If $N$ is normal in $G$, then $P \cap N$ is a Sylow $p$-subgrup of $N$. But if $N$ is not normal in $G$ , there is also the issue? ...
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1answer
66 views

$G$ be a finite simple group , then every element of $G$ can be written as a product of $n$-th powers of elements of $G$?

Let $G$ be a finite simple group , let $n$ be a positive integer such that not all $n$-th powers of elements of $G$ are identity , then is it true that every element of $G$ can be written as a ...
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3answers
100 views

Property of odd ordered elements of a Group

I'm (slowly) working my way through "Abstract Algebra" by Dummit and Foote. In the first set of exercises on group theory, the following question is posed: "Let $G$ be a finite group and let $x$ be ...
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1answer
49 views

Finite {2.3}-group with 4 Sylow 3-subgroup

Let $G$ be a finite {$2$,$3$}-group, the number of Sylow $3$-subgroups of $G$ be $4$, and a Sylow $2$-subgroup of $G$ be normal in $G$. Let $N$ be the kernel of the conjugation action of $G$ on its ...
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2answers
48 views

If $\sigma \in S_n$ has order some prime $p$, then is $|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p$? [closed]

Let $\sigma \in S_n$ be such that $o(\sigma)=p$ (some prime). Then is it true that $$|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p\ ?$$
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1answer
67 views

Name of the theorem: If $p^k$ divides $|G|$, then $G$ has a subgroup of order $p^k$?

Note: I am not asking for a proof of this theorem or any other theorem or help with a mathematical problem. This question is a reference request. I use the following well-known and somewhat-easy-to-...
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1answer
70 views

Questions about $\mathrm{SL}_2(\mathbb{F}_7)$

Let $G=\mathrm{SL}_2(\mathbb{F}_7)$, which has order $336=2^4\cdot 3\cdot 7$. And I may assume that $G$ is generated by the two matrices $$\begin{pmatrix}1&1\\0&1\end{pmatrix}, \begin{pmatrix}...
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1answer
38 views

Image of Sylow $p$-subgroup is Sylow $p$-subgroup

If $f:G\to H$ is an epimorphism between finite groups and $K\subset G$ a Sylow p-group then I want to show that $f(K)\subset H$ is also a Sylow p-group. So we want to show that $|f(K)|=p^m$ where $m$...
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461 views

Is it a subgroup?!

Let $G$ be a finite group, $A$ an its subset and put $A^{-1}=\{ a^{-1}:a\in A\}$. Is it true that if $A$ is symmetric (i.e., $A=A^{-1}$), $G=AB$ and $|AB|=|A||B|$, for some $B\subseteq G$, then $A$ ...
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1answer
65 views

Showing a polynomial of degree 7 is not solvable by radicals. [closed]

Show that the polynomial of $x^7-10x^5+15x+5$ is not solvable by radicals. For a polynomial of degree 5, we simply consider the derivative and determine the number of real and complex roots from ...
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30 views

Is there a simple way to find the conjugacy classes of $A_n$? [duplicate]

For the symmetric groups $S_n$ there's a quite simple way to find all the conjugacy classes. We find partitions of $n$, that is, we write $$n = n_1+\cdots+n_k,$$ and then for each such partition ...
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3answers
28 views

Let $G=\langle \mathbb{Z},+\rangle $ and $H=\{6n|n \in \mathbb{Z}\}$. Find all the distinct left and right cosets of $H$ in $G$.

I have an exercise where I am supposed to find the left and right cosets. But how do I generate the cosets? As I have understood it you are supposed to pick a number that is not in the set $H$ and ...
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171 views

Is there an interval of finite groups, at index n, with strictly more elements than the subgroup lattice of any group, of order n? [migrated]

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$. There is an OEIS page for the sequence $s(n)$: A018216 1, 2, 2, 5, 2, ...
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2answers
95 views

Counting Circular Sequence (Burnside Lemma?)

How many distinct circular binary sequences of length $n$ are there? How many distinct circular binary sequences of length $n$ containing a given pattern, e.g., $110$ are there? The same questions as ...
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1answer
49 views

Minimal Generating Set of Group $G$ with size $k$

$G$ is a permutation group. So, $G < S_n$ where $S_n$ is a symmetric group acts on $n$ object. $G$ is not isomorphic to any symmetric or alternating group, i.e. $G \neq S_t , A_t$ for $1 <t \...
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1answer
28 views

multiple of a crossed homomorphism from finite group to a divisible one is principal

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed ...
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41 views

Characteristics of group $G$ which is not a Symmetric Group or an Alternating Group.

$G$ is a permutation subgroup of symmetric group $S_n$ such that- $G \neq S_k$ or $A_k$ for $1 \leq k < n $. In other words, $G$ is not a symmetric group or an alternating group. What are the ...
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1answer
24 views

Example of a group $G$ that has a $\pi$-subgroup $U$ which isn't contained in any $\pi$ Hall subgroup

I would like to know if it is possible to find a group $G$ such that: $G$ has a $\pi$-subgroup $U$ but $G$ has no $H$ $\pi$-Hall subgroup such that $U$ is contained in $H$, where $\pi$ is a set of ...
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1answer
56 views

Order of a Subgroup

Let, $A \subset S_n$, $S_n$ is a symmetric group. $|A| \leq \log (n!)$. $A$ generates a subgroup $G$ of $S_n$. i.e. $\langle A \rangle=G < S_n$. What is the order of $G$? Can it be bounded by $|...
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1answer
33 views

Recognizing Permutation of Group with different Label

Problem description: Assume, a group, $G \leq S_{26}$ , $S_{26}$ is a symmetric group. Each permutation of $G$ is labeled using $1,2,....26$ as usual. Suppose, $f$ is a function that changes label ...
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1answer
53 views

Do arbitrary $\log_2(|G|)$ elements generate a group?

A set $S$ has $\log_2(|G|)$ distinct elements (arbitrary) of permutation group $G \leq S_n$. $S_n$ is symmetric group. i.e. $S \subset G$ and $|S| = \log_2(|G|) $. Is it a generating set of $G$? i....
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Spherical functions which are invariant under a finite rotation group

Is there a nice, clean reference which lists a basis in terms of (linear combinations of) spherical harmonics for the $L^2$ space of functions defined on the sphere $\mathbb{S}^2$ which are invariant ...
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28 views

Showing $PSL(2,9)$ doesn't have a subgroup of order $90$

I got stuck in proving that there is not any subgroup of order $90$ in group $PSL(2,9) = SL(2,9)/Z(SL(2,9))$. Can anyone help?
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38 views

Generating set of same size must have common elements

Generating set $S_1, S_2$ generates permutation group $G$ where the number of elements in $S_1$ is equal to the number of elements in $S_2$. Prove, $S_1 \cap S_2 \neq \emptyset $ (not considering ...
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31 views

Automorphism and Direct Product of Generating Set

Notation: $H $ are partitioned into sub-graphs $ H_1,H_2 \cdots H_x$ . We see them in the adjacency matrix of $H$ given below- $$H = \begin{bmatrix} H_{(x)} & R_{(x, x-1)} & R_{(x,x-...
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21 views

How can I prove that $PSL(2,3)\simeq A_4$?

I would like to prove that $PSL(2,3) \simeq A_4$. I know the structure of $SL(2,3)$ and that $A_4=V\rtimes C_3$, where $V$ is the Klein subgroup. How can I proceed? Thanks for the help!
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On representations of a nonabelian group of order $pq$

Let $p,q$ primes number s.t. $p>q$ and let $G$ a non abelian group of order $pq$. 1) Determine all degree of irreducible representation 2) Show that $|[G,G]|=p$ (where $[G,G]=\left<ghg^...
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1answer
39 views

Showing a surjective homomorphism maps Sylow $p$-groups to Sylow $p$-groups [duplicate]

If $f:G\to H$ is a surjective homomorphism of finite groups, then $f$ sends Sylow $p$-subgroups to Sylow $p$-subgroups. Here's what I have. Suppose $\vert G \vert=p^km$ with $(p,m)=1$. Let $P\in \...
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4answers
65 views

Number of homomorphisms between two cyclic groups.

Is it true that the number of homomorphisms between any two finite cyclic groups of order $m\,\&\,n$ is $\gcd(m,n)$? I have posted an answer which I believe is true, just wanted to know different ...
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1answer
21 views

Solvability and representation of finite groups

Let $G$ be a finite solvable group, and let $G=G^{(0)}\unrhd G^{(1)}\unrhd...G^{(n)}=1$ be its derived series. Is it true that any irreducible representation of $G$ has dimension at most $n$?
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22 views

Non-Isomorphic Groups generated by a Set of fixed cardinaity

Given a set of permutations $A \subset S_n$. It has $|A|$ (the cardinality of $A$) elements. We can construct a group using $A$. How many non-isomorphic groups(all with the same order) can we ...
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45 views

Compute the Galois group of $p(x)=x^4+ax^3+bx^2+cx+d$

Compute the Galois group of the following polynomial: $$p(x)=x^4+ax^3+bx^2+cx+d$$ Step 1: Calculate cubic resolvent Step 2: Calculate the discrimimant $\gamma$ of the cubic resolvent Step 3: If $...
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1answer
67 views

Why all irreducible representations appear in the regular representation?

Let $G$ be a finite group and $R$ the regular representation. That is, as a vector space $R = F(G)$ is the free vector space with basis $G$. If the basis is $\{e_g : g \in G\}$ the action is defined ...
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1answer
24 views

Local group rings

Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
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65 views

Decomposition of a representation into a direct sum of irreducible ones

I'm studying representation theory and in the book (Fulton and Harris) the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite ...
3
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2answers
97 views

$(C_2 )^3$ is not a subgroup of $S_4$

Prove $(C_2)^3$ is not a subgroup of $S_4$. (Using group actions.) I could think of a permutation argument that $(C_2)^3$ is not a subgroup of $S_4$. But I would like to argue it by considering ...
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1answer
24 views

Problems in understanding a passage in the proof of Grün theorem for transfer

This is the statement of the theorem: Let $P$ a Sylow $p$-subgroup of $G$ and $Z$ a subgroup of $Z(P)$ that is weakly closed in $P$. Set $H=N_G(Z)$. Then $P\cap G'=P\cap H'$ and $P/(P\cap G')\simeq ...
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13 views

Isomorphism classes and invariant factors of abelian group

Let $G$ be an abelian group with $ord(G)=3374=2\cdot 7\cdot 241$. Calculate all isomorphism classes with the invariant factors $k_1\ ...k_n$ sucht that $k_i$ divides $k_j$ $(i<j)$. Since $ord(G)=...
2
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1answer
50 views

Galois Theory.Subgroups of Galois Group

How to List the subgroups of the Galois group in general.Im not interested in a specific Example but to make it easier. Supposes the galois group $G=Gal[Q(v,i):Q]$ $$v= \sqrt[4]{2}$$ I know how to ...